Liouville theorem,conformally invariant cones and umbilical surfaces for Grushin-type metrics |
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Authors: | Daniele Morbidelli |
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Institution: | 1.Dipartimento di Matematica,Università di Bologna,Bologna,Italy |
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Abstract: | We prove a classification theorem for conformal maps with respect to the control distance generated by a system of diagonal
vector fields in ℝ
n
. It turns out that in many cases all such maps can be obtained as compositions of suitable dilations, inversions and isometries.
Our methods involve a study of the singular Riemannian metric associated with the vector fields. In particular, we identify
some conformally invariant cones related to the Weyl tensor. The knowledge of such cones enables us to classify all umbilical
hypersurfaces. |
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Keywords: | |
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